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<title>Creating Permutation Matrices - GNU Octave</title>
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<h4 class="subsection">21.1.2 Creating Permutation Matrices</h4>
<p>For creating permutation matrices, Octave does not introduce a new function, but
rather overrides an existing syntax: permutation matrices can be conveniently
created by indexing an identity matrix by permutation vectors.
That is, if <var>q</var> is a permutation vector of length <var>n</var>, the expression
<pre class="example"> P = eye (n) (:, q);
</pre>
<p class="noindent">will create a permutation matrix - a special matrix object.
<pre class="example"> eye (n) (q, :)
</pre>
<p class="noindent">will also work (and create a row permutation matrix), as well as
<pre class="example"> eye (n) (q1, q2).
</pre>
<p>For example:
<pre class="example"> eye (4) ([1,3,2,4],:)
⇒
Permutation Matrix
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
eye (4) (:,[1,3,2,4])
⇒
Permutation Matrix
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
</pre>
<p>Mathematically, an identity matrix is both diagonal and permutation matrix.
In Octave, <code>eye (n)</code> returns a diagonal matrix, because a matrix
can only have one class. You can convert this diagonal matrix to a permutation
matrix by indexing it by an identity permutation, as shown below.
This is a special property of the identity matrix; indexing other diagonal
matrices generally produces a full matrix.
<pre class="example"> eye (3)
⇒
Diagonal Matrix
1 0 0
0 1 0
0 0 1
eye(3)(1:3,:)
⇒
Permutation Matrix
1 0 0
0 1 0
0 0 1
</pre>
<p>Some other built-in functions can also return permutation matrices. Examples
include
<dfn>inv</dfn> or <dfn>lu</dfn>.
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